Computing the fourth RV moment did require to recourse to the Weingarten calculus, whose apparatus, arguably demanding for the neophyte, turned out decisive for the pursuit of our objective. One may reasonably hope that future developments along that line will enrich the present results, replacing in particular the mechanical computation of Tables REF  and REF  by true mathematical arguments. However, determining the analytical, exact null distribution of \({\tt RV}\) , may reveal itself out of reach: as a matter of fact, the moment generating function (REF ) is an orthogonal analog of the celebrated Harish-Chandra trace integral for the unitary group, whose analytical expression has been determined ever since the fifties [1]} (see also e.g. [2]} and [3]}). Yet, discovering a corresponding expression for the orthogonal case, precisely, has not been achieved so far.
